Normal Incidence Excitation of Out-of-Plane Lattice Resonances in Bipartite Arrays of Metallic Nanostructures

As a result of their coherent interaction, two-dimensional periodic arrays of metallic nanostructures support collective modes commonly known as lattice resonances. Among them, out-of-plane lattice resonances, for which the nanostructures are polarized in the direction perpendicular to the array, are particularly interesting since their unique configuration minimizes radiative losses. Consequently, these modes present extremely high quality factors and field enhancements that make them ideal for a wide range of applications. However, for the same reasons, their excitation is very challenging and has only been achieved at oblique incidence, which adds a layer of complexity to experiments and poses some limitations on their usage. Here, we present an approach to excite out-of-plane lattice resonances in bipartite arrays under normal incidence. Our method is based on exploiting the electric-magnetic coupling between the nanostructures, which has been traditionally neglected in the characterization of arrays made of metallic nanostructures. Using a rigorous coupled dipole model, we demonstrate that this coupling provides a general mechanism to excite out-of-plane lattice resonances under normal incidence conditions. We complete our study with a comprehensive analysis of a potential implementation of our results using an array of nanodisks with the inclusion of a substrate and a coating. This work provides an efficient approach for the excitation of out-of-plane lattice resonances at normal incidence, thus paving the way for the leverage of the extraordinary properties of these optical modes in a wide range of applications.


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Absorbance spectrum (upper panel) and out-of-plane magnetic dipole spectra for each of the two particles in the unit cell (lower panel).(b) Reflectance spectrum (upper panel) and in-plane electric dipole spectrum for both particles in the unit cell (lower panel).For comparison, the dashed curves in the absorbance and reflectance spectra show results obtained from FEM simulations.as depicted in Figure S3.Consequently, the effective polarizability of the array A, defined in Equation 2 of the main paper, has to be invariant under such transformations.This is expressed mathematically by where Π z , Π u , and Π v represent the reflection operators with respect to the plane uv, vz, and uz, respectively.Using the same notation as in the main paper, these matrices are defined as and Notice that the negative signs in these matrices are a consequence of the pseudovector nature of the magnetic dipole and field.Furthermore, σ z , σ u , and σ v are 3 × 3 matrices, which in the basis formed by the vectors û, v, and ẑ, read .

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Figure S1: Spectra of the real part of the inverse of α EE and of the in-plane and outof-plane components of Re{G EE 11 } (upper panel).Spectra of the in-plane and out-of-plane components of the electric dipole for the illumination conditions of Figure 2 of the main paper (lower panel).The vertical dotted lines indicate the maxima of the different electric dipole components.

Figure S2 :
Figure S2: Optical response of the array analyzed in Figure 2 when excited with the orthogonal polarizationE i = (1, −1, 0)E 0 / √ 2 and H i = (1, 1, 0)E 0 / √ 2. (a)Absorbance spectrum (upper panel) and out-of-plane magnetic dipole spectra for each of the two particles in the unit cell (lower panel).(b) Reflectance spectrum (upper panel) and in-plane electric dipole spectrum for both particles in the unit cell (lower panel).For comparison, the dashed curves in the absorbance and reflectance spectra show results obtained from FEM simulations.

Figure S3 :
Figure S3: Schematics of the symmetry planes of the bipartite array of Figure 2 of the main paper.
S1 imposes that the EE and MM terms of the effective polarizability of the array are diagonal, while for the EM and ME terms, only the components uz and zu take finite values.This is in perfect accordance with the results of Figure 4(b) of the main paper.Moreover, Equation S1 also implies that A ς 11 = −A ς 22 and A ς 12 = −A ς 21 for ς = EM and ς = ME, which is consistent with p 1,z and p 2,z having opposite signs.

Figure S5 :
Figure S5: (a) Scheme of the unit cell of the triangular array.(b) Relevant terms of the lattice sum G 21 calculated at the first Rayleigh anomaly (λ = √ 3a/2) and normalized to Re{G EE 11,xx }, as a function of r 21 .(c) Absorbance spectrum (upper panel) and out-of-plane electric dipole spectra for each of the two particles in the unit cell (lower panel).(d) Reflectance spectrum (upper panel) and in-plane electric dipole spectrum for both particles in the unit cell (lower panel).For comparison, the dashed curves in the absorbance and reflectance spectra show results obtained from FEM simulations.